Theorem sseqtrrd 3181

Description: Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)

Hypotheses

Ref	Expression
sseqtrrd.1	|- ( ph -> A C_ B )
sseqtrrd.2	|- ( ph -> C = B )

Assertion

Ref	Expression
sseqtrrd	|- ( ph -> A C_ C )

Proof of Theorem sseqtrrd

Step	Hyp	Ref	Expression
1		sseqtrrd.1	. 2 |- ( ph -> A C_ B )
2		sseqtrrd.2	. . 3 |- ( ph -> C = B )
3	2	eqcomd 2171	. 2 |- ( ph -> B = C )
4	1, 3	sseqtrd 3180	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1343   C_ wss 3116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-in 3122  df-ss 3129

This theorem is referenced by:  sseqtrrid  3193  fnfvima  5719  tfrlemiubacc  6298  tfr1onlemubacc  6314  tfrcllemubacc  6327  rdgivallem  6349  nnnninf  7090  dfphi2  12152  ctinf  12363  toponss  12664  ssntr  12762  iscnp3  12843  cnprcl2k  12846  tgcn  12848  tgcnp  12849  ssidcn  12850  cncnp  12870  txcnp  12911  imasnopn  12939  hmeontr  12953  blssec  13078  blssopn  13125  xmettx  13150  metcnp  13152  nnsf  13885  nninfsellemsuc  13892